Effective Hamiltonians for motions with disparate time scales: The quantum shell model and the classical statistical shell model

The shell model of atomic polarizability is generalized to include a statistical distribution of shell states for each instantaneous configuration of the cores. Effective Hamiltonians for motions of the cores are derived for both quantum and classical descriptions of the shells. This extends the shell model into a formalism useful for molecular -dynamics. The fluctuations of the shell positions account for London dispersion interactions. The resulting equations may also have applications to other systems containing motions with disparate time scales.